Pairs of SAT Assignments and Clustering in Random Boolean Formulæ

We investigate geometrical properties of the random K-satisfiability problem. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where SAT assignments are grouped into well separated clusters. This confirms the validity of the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method relies on the study of pairs of SAT-assignments at a fixed distance. It uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.